Positive solutions of positive linear systems
The Volterra-Lotka equations \(dx_ i/dt=x_ i(b_ i-\Sigma a_{ij}x_ j),\) \(b_ i>0\), \(a_{ii}>0\), \(a_{ij}\geq 0\) correspond to an ecosystem with interspecies and intraspecies competition. A steady state solution \(X^*\) exists if \(b_ i-\Sigma a_{ij}X^*_ j=0\). The author shows that A is invertible with \(A^{-1}b\) having all positive elements, so that such a stable \(X^*\) exists, provided \[ b_ i>\sum^{n}_{j=1,j\neq i}a_{ij}b_ j/a_{jj},\quad i=1,...,n. \] An iterative method of approximating \(X^*\) is given.
scientific article; zbMATH DE number 3901176